Good metric on $C^k(0,1)$ and $C^\infty(0,1)$

Ask yourself if the spaces $C^k(0,1)$ are of interest.

EDIT: I also suggest thinking more about and therefore formalising the notion of a 'good' metric on a space. You might find that a good metric is one that induces a topology on the space that has desirable properties for doing analysis, like local convexity, being complete, the Heine-Borel property, an interesting dual space, etc.

Since $(0,1)$ is the union of countably many compact sets $K_n$ such that $K_n\subset\text{int}K_{n+1}$, then we define $C(0,1)$ as the space of all continuous real-valued functions on $(0,1)$ with topology induced by the family of separating semi-norms: $$ p_n(f):=\sup_{x\in K_n}|f(x)|. $$ Then the sets $$ V_n:=\{f\in C(0,1)\;|\;p_n(f)<1/n\},\qquad n=1,2,... $$ form a local convex base for $C(0,1)$. This topology is in fact induced by the metric $$ d(f,g):=\max_n\frac{1}{2^n}\frac{p_n(f-g)}{1+p_n(f-g)} $$ This metric is complete, so $C(0,1)$ becomes a Frechet space.

The space $C^\infty(0,1)$ is defined to be the space of all real-valued functions $f$ on $(0,1)$ with the property that $D^kf\in C(0,1)$ for all $k\in\mathbb{N}_0$. Choosing the same compact sets $K_n$ as above, the following family of semi-norms undices a metrizable, locally-convex topology on $C^\infty(0,1)$: $$ p_n(f):=\max_{x\in K_n,\;k\leq n}|D^kf(x)|. $$ The metric on this space is of the same form as above. It can be shown that this space is also a Frechet space and additionally has the Heine-Borel property, so it follows that it is not normable (the metric is not induced by a norm). On the space $C^\infty(0,1)$ is constructed the space of distributions with compact support, which are linear functionals that are continuous with respect to the above defined topology.

To generalise the above, you should replace $(0,1)$ with an arbitrary open subset of $\Omega\subset\mathbb{R}^m$, and allow $k$ to be a multi-index.

Other continuous function spaces that you might be interested in are:

(i) the space $\mathcal{D}_K$ which is the space of all $f\in C^\infty(\mathbb{R}^m)$ such that $\text{supp}f\subset K$ where $K\subset\Omega$ is compact. It can be shown that $\mathcal{D}_K$ is a closed subspace of $C^\infty(\Omega)$.

(i) the Schwartz space $\mathcal{S}(\mathbb{R}^m)$, which is important for Fourier analysis. On this space is consructed the space of tempered distributions, which is composed of linear functionals that are continuous with respect to the appropriate topology defined on the Schwartz space.

(ii) the space of test functions $\mathcal{D}(\Omega)=C^\infty_0(\Omega)$, which is important for Sobolev space and PDE theory. The space of distributions is constructed on this space, and consists of all linear functionals that are continuous with respect to the appropriate topology on $\mathcal{D}(\Omega)$. This topology is a challenge to define and understand, but is easily characterised in terms of convergence.

By far the best reference for all of this stuff is Rudin's book on functional analysis. Also see Yosida's book on functional analysis, DiBenedetto's book on real analysis and Knapp's book on advanced analysis.

We can define the metrics $$d(f,g):=\sum_{n=1}^{+\infty}2^{-n}\min\left\{1,\max_{0\leqslant k\leqslant n}\max_{n^{-1}\leqslant x\leqslant 1-n^{-1}}|f^{(k)}(x)-g^{(k)}(x)|\right\}\quad\mbox{on } C^\infty(0,1)$$ $$d_N(f,g):=\sum_{n=1}^{+\infty}2^{-n}\min\left\{1,\max_{0\leqslant k\leqslant N}\max_{n^{-1}\leqslant x\leqslant 1-n^{-1}}|f^{(k)}(x)-g^{(k)}(x)|\right\}\quad\mbox{on } C^N(0,1).$$ This gives information about the behavior of the functions and their derivatives on compact intervals. Summing this help us to know the behavior on $(0,1)$. Furthermore, one can check that $C^\infty(0,1)$ and $C^N(0,1)$ endowed with the corresponding metrics are complete.

Note that I just used the fact that the natural topologies on $C^k(0,1)$ and $C^\infty(0,1)$ are generated by a countable family of semi-norms.

We can give a generalization when we replace $(0,1)$ by any open subset of $\Bbb R^d$, $d\geqslant 1$.

The space $C^k([a,b])$ is a normed space for each $k$ and for each pair $a < b$ of real numbers with the norm

$\|f\|_{C^0} = \sup_{x\in[a,b]} |f(x)|$ for $k=0$

and

$\|f\|_{C^k} = \sum_{|s|\le k} \|\partial^sf\|_{C^0}$ for $k>1$

(For the $C^0$-case, it is important that the interval is closed, as there are continuous functions that live on $(a,b)$ but are not bounded (i.e. they run towards $\infty$ if $x\to a$ or $x\to b$).)

Proving the norm axioms for the $C^0$ case is not trivial but should be contained in textbooks on functional analysis. With this, proving the norm axioms for the $C^k$-case is trivial.

Now, every norm induces a metric: $d(f,g) := \|f-g\|_{C^k([a,b])}$.

Also, these norms make $C^k$ a Banach space (each Cauchy sequence converges in the space itself). A proof for this statement should also be contained in functional analysis textbooks.

Hope this helps you a bit.

P.S.: A good functional analysis textbook in German would be "Hans Wilhelm Alt: Lineare Funktionalanalysis" in my opinion. All the proofs I hinted at above and much more can be found there.

P.P.S: To address some more of your questions: Generally, the notion of a norm is 'better' (as in more good ;) ) than just a metric, since it gives you some knowledge of the 'length' or 'size' of elements, not just the distance between them. As mentioned above, this is more general.

In the finite dimensional case, one can show that every norm is equivalent, which means that they induce the same topology (with every metric you can define 'open balls', which are a basis of a topology). This statement is known as the Heine-Borel theorem.

However, in the infinite dimensional case (like $C^k$), there are different norms that are not equivalent. The norm for $C^k([a,b])$ I stated above is, however, the generally most used norm on these spaces and should be sufficient for basic linear functional analysis - as far as I know.

I cannot say if this is the 'best' metric, but this is a widely used one. For everything I just wrote, the book I mentioned above is a good reference especially in the first chapters where topology, norms and metrics are treated.